A Leakage-Resilient Ciphertext-Policy Attribute-based Encryption (LR-CP-ABE) not only supports the fine-grained access control to encrypted data but also guarantees the security of the data under the side-channel attacks. However, the leakage rate in the existing schemes is low or related to the number of attributes. It will make these schemes suffer from continual attacks. In addition, all of them almost not consider the leakage of the users' privacy and rely on the composite order groups which will threaten the privacy security of the users and depress the users in practice. In this paper, we aim at solving the above problems and propose a scheme with the improving leakage rate in the prime order group. In the proposed scheme, an extension of the lattice-based trapdoor is used to make it achieve the maximum leakage rate 1-o(1). Moreover, it achieves the anonymity which can protect the privacy of the receivers. The proposed scheme can be reduced to the standard assumption-Decision Linear (DLIN) assumption in the selective security model and resist the Chosen Plaintext Attacks (CPA security). At last, the performance comparisons are given to confirm the efficiency and security of the proposed scheme.